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{\it DGM}

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Mahboub Baccouch\\
Virginia Tech\\
{\tt baccouch@vt.edu}
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\noindent{\bf Abstract}:  In the first part of this presentation we
investigate higher-order discontinuous Galerkin methods (DGM) for hyperbolic
problems on triangular meshes. A detailed description of the formulation of
the DGM on a hyperbolic partial differential equation is presented. We study
the effect of finite element spaces on the superconvergence properties of DG
solutions on three types of triangular elements. We show that the DG solution
is $O(h^{p+2})$ superconvergent at Legendre points on the outflow edge on
triangles having one outflow edge using three polynomial spaces. For triangles
having two outflow edges the finite element error is $O(h^{p+2})$
superconvergent at the end points of the inflow edge for an augmented space
$\mathcal{U}_p$. Furthermore, we discovered additional supeconveregence points
in the interior of triangles.  We also established a global superconvergence
result on meshes consisting of triangles of type III only. Several numerical
simulations are performed to validate the theory.  Superconvergence on more
general meshes as well as {\it a posteriori} error estimates based on these
superconvergence results are presented in the second part of this
presentation. We apply these superconvergence results to construct simple,
efficient and asymptotically correct {\it a posteriori} error estimates for
discontinuous finite element solutions of scalar first-order hyperbolic
partial differential problems on triangular meshes in smooth solution
regions.  Based to these superconvergence results, we explicitly write the
basis functions for the leading term of the error corresponding to several
finite element spaces and types of elements and construct asymptotically
correct {\it a posteriori} error estimates by solving local hyperbolic
problems with no boundary conditions on more general meshes. The leading term
of the discretization error on each triangle is estimated by solving a local
problems where no boundary conditions are needed. The computed error estimates
are shown to converge to the true error under mesh refinement in smooth
solution regions. We also show global superconvergence for discontinuous
solutions on general unstructured meshes. The error estimates are tested on
several problems, both linear and nonlinear, to show their efficiency,
accuracy and asymptotic convergence of the error estimates under mesh
refinement.

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